The Memorization Problem: Softmax reaches only the interior of the simplex

Lopez-Lira, Tang & Zhu (2025) · arXiv:2504.14765 (v2) · summary on the ideas page

Appendix C.3 claims finite real logits realize every distribution on a finite label set. That fails at the boundary. The error is local and repairable, and the paper's central non-identification result does not depend on it.

The claim

Appendix C.3's stochastic extension states that finite real logits can realize every probability distribution over a finite label set.

For finite z, every softmax coordinate is strictly positive:

softmax(z)_i = exp(z_i) / sum_j exp(z_j) > 0 softmax([50, 0]) = (1.000000000000000, 1.93e-22) softmax([200, 0]) = (1.000000000000000, 1.38e-87) softmax([700, 0]) = (1.000000000000000, 9.86e-305)

So (1, 0) has no finite-logit representation. Interior distributions do, by z_i = log p_i; boundary distributions are reachable only as limits, or with explicit masking or infinite logits.

What survives

Almost all of it. Restrict exact reachability to the interior, or state the result as an approximation, and the extension goes through. The deterministic observational-equivalence construction does not use this step, and the non-identification argument stands. That argument does assume an unrestricted relation between factual and counterfactual parameters, so it shows that skill and memorisation cannot be separated without further structure, not that every additional assumption or intervention is useless.

Mathematical error, repairable

Appendix C.3 incorrectly claims finite-logit softmax is onto the closed simplex. This is a minor repairable error. The core non-identification result survives and should be read with its unrestricted-model assumptions attached.

Appendix C.3. Softmax positivity checked numerically at large logit gaps. Numerical checks are reproducible with reviews/counterexamples.py (standard library only). These are counterexamples to particular claims; they do not reproduce the papers' training experiments.

← TEMPO · Temporal Leakage in LLM Backtesting →